text: The Jacobi and Dedekind ellipsoids are both equilibrium figures for a body of rotating homogeneous self-gravitating fluid. However, while the Jacobi ellipsoid spins bodily, with no internal flow of the fluid in the rotating frame, the Dedekind ellipsoid maintains a fixed orientation, with the constituent fluid circulating within it. This is a direct consequence of Dedekind's theorem.
  For any given Jacobi ellipsoid, there exists a Dedekind ellipsoid with the same semi-principal axes  and same mass and with a flow velocity field of
  where  are Cartesian coordinates on axes  aligned respectively with the  axes of the ellipsoid. Here  is the vorticity, which is uniform throughout the spheroid (). The angular velocity  of the Jacobi ellipsoid and vorticity of the corresponding Dedekind ellipsoid are related by
  That is, each particle of the fluid of the Dedekind ellipsoid describes a similar elliptical circuit in the same period in which the Jacobi spheroid performs one rotation.
  In the special case of , the Jacobi and Dedekind ellipsoids (and the Maclaurin spheroid) become one and the same; bodily rotation and circular flow amount to the same thing. In this case , as is always true for a rigidly rotating body.
  In the general case, the Jacobi and Dedekind ellipsoids have the same energy, but the angular momentum of the Jacobi spheroid is the greater by a factor of